Mary,
My grandfather was Jon Vernhardson the younger; he remained "Vernhardson" until his death in 1934. The elder Jon became "Leifur" for reasons unknown to me. I have much more info--have contacted one daughter and grand-daughter of Jon the elder, and many of the Sturlaugson branch.
Richard E. Johnson, Jr.
(360) 456-0567 or rjonasenhus@attbi.com